This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License without attribution as requested by the work’s original creator or licensee. Preface Introduction and Background
Quantification of Uncertainty via Probability Models
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- A Historical Prelude to the Quantification of Uncertainty Via Probabilities
- Measurement Techniques for Frequency, Severity, and Probability Distribution Measures for Quantifying Uncertain Events
- Measures of Frequency and Severity
- Frequency and Probability
2.1 Quantification of Uncertainty via Probability Models LEARNING OBJECTIVES
As we consider uncertainty, we use rigorous quantitative studies of chance, the recognition of its empirical regularity in uncertain situations. Many of these methods are used to quantify the occurrence of uncertain events that represent intellectual milestones. As we create models based upon probability and statistics, you will likely recognize that probability and statistics touch nearly every field of study today. As we have internalized the predictive regularity of repeated chance events, our entire worldview has changed. For example, we have convinced ourselves of the odds of getting heads in a coin flip so much that it’s hard to imagine otherwise. We’re used to seeing statements such as “average life of 1,000 hours” on a package of light bulbs. We understand such a phrase because we can think of the length of life of a light bulb as being uncertain but statistically predictable. We routinely hear such statements as “The chance of rain tomorrow is 20 percent.” It’s hard for us to imagine that only a few centuries ago people did not believe even in the existence of chance occurrences or random events or in accidents, much less explore any method of quantifying seemingly chance events. Up until very recently, people have believed that God controlled every minute detail of the universe. This belief rules out any kind of conceptualization of chance as a regular or predictable phenomenon. For example, until recently the cost of buying a life annuity that paid buyers $100 per month for life was the same for a thirty-year-old as it was for a seventy-year-old. It didn’t matter that empirically, the “life expectancy” of a thirty-year-old was four times longer than that of a seventy-year-old. [1] After all, people believed that a person’s particular time of death was “God’s will.” No one believed that the length of someone’s life could be judged or predicted statistically by any noticed or exhibited regularity across people. In spite of the advancements in mathematics and science since the beginning of civilization, remarkably, the development of measures of relative frequency of occurrence of uncertain events did not occur until the 1600s. This birth of the “modern” ideas of chance occurred when a problem was posed to mathematician Blaisé Pascal by a frequent gambler. As often occurs, the problem turned out to be less important in the long run than the solution developed to solve the problem. The problem posed was: If two people are gambling and the game is interrupted and discontinued before either one of the two has won, what is a fair way to split the pot of money on the table? Clearly the person ahead at that time had a better chance of winning the game and should have gotten more. The player in the lead would receive the larger portion of the pot of money. However, the person losing could come from behind and win. It could happen and such a possibility should not be excluded. How should the pot be split fairly? Pascal formulated an approach to this problem and, in a series of letters with Pierre de Fermat, developed an approach to the problem that entailed writing down all possible outcomes that could possibly occur and then counting the number of times the first gambler won. The proportion of times that the first gambler won (calculated as the number of times the gambler won divided by the total number of possible outcomes) was taken to be the proportion of the pot that the first gambler could fairly claim. In the process of formulating this solution, Pascal and Fermat more generally developed a framework to quantify the relative frequency of uncertain outcomes, which is now known as probability. They created the mathematical notion of expected value of an uncertain event. They were the first to model the exhibited regularity of chance or uncertain events and apply it to solve a practical problem. In fact, their solution pointed to many other potential applications to problems in law, economics, and other fields. From Pascal and Fermat’s work, it became clear that to manage future risks under uncertainty, we need to have some idea about not only the possible outcomes or states of the world but also how likely each outcome is to occur. We need a model, or in other words, a symbolic representation of the possible outcomes and their likelihoods or relative frequencies. A Historical Prelude to the Quantification of Uncertainty Via Probabilities Historically, the development of measures of chance (probability) only began in the mid-1600s. Why in the middle ages, and not with the Greeks? The answer, in part, is that the Greeks and their predecessors did not have the mathematical concepts. Nor, more importantly, did the Greeks have the psychological perspective to even contemplate these notions, much less develop them into a cogent theory capable of reproduction and expansion. First, the Greeks did not have the mathematical notational system necessary to contemplate a formal approach to risk. They lacked, for example, the simple and complete symbolic system including a zero and an equal sign useful for computation, a contribution that was subsequently developed by the Arabs and later adopted by the Western world. The use of Roman numerals might have been sufficient for counting, and perhaps sufficient for geometry, but certainly it was not conducive to complex calculations. The equal sign was not in common use until the late middle ages. Imagine doing calculations (even such simple computations as dividing fractions or solving an equation) in Roman numerals without an equal sign, a zero element, or a decimal point! But mathematicians and scientists settled these impediments a thousand years before the advent of probability. Why did risk analysis not emerge with the advent of a more complete numbering system just as sophisticated calculations in astronomy, engineering, and physics did? The answer is more psychological than mathematical and goes to the heart of why we consider risk as both a psychological and a numerical concept in this book. To the Greeks (and to the millennia of others who followed them), the heavens, divinely created, were believed to be static and perfect and governed by regularity and rules of perfection—circles, spheres, the six perfect geometric solids, and so forth. The earthly sphere, on the other hand, was the source of imperfection and chaos. The Greeks accepted that they would find no sense in studying the chaotic events of Earth. The ancient Greeks found the path to truth in contemplating the perfection of the heavens and other perfect unspoiled or uncorrupted entities. Why would a god (or gods) powerful enough to know and create everything intentionally create a world using a less than perfect model? The Greeks, and others who followed, believed pure reasoning, not empirical, observation would lead to knowledge. Studying regularity in the chaotic earthly sphere was worst than a futile waste of time; it distracted attention from important contemplations actually likely to impart true knowledge. It took a radical change in mindset to start to contemplate regularity in events in the earthly domain. We are all creatures of our age, and we could not pose the necessary questions to develop a theory of probability and risk until we shook off these shackles of the mind. Until the age of reason, when church reforms and a growing merchant class (who pragmatically examined and counted things empirically) created a tremendous growth in trade, we remained trapped in the old ways of thinking. As long as society was static and stationary, with villages this year being essentially the same as they were last year or a decade or century before, there was little need to pose or solve these problems. M. G. Kendall captures this succinctly when he noted that “mathematics never leads thought, but only expresses it.”* The western world was simply not yet ready to try to quantify risk or event likelihood (probability) or to contemplate uncertainty. If all things are believed to be governed by an omnipotent god, then regularity is not to be trusted, perhaps it can even be considered deceptive, and variation is irrelevant and illusive, being merely reflective of God’s will. Moreover, the fact that things like dice and drawing of lots were simultaneously used by magicians, by gamblers, and by religious figures for divination did not provide any impetus toward looking for regularity in earthly endeavors. * M. G. Kendall, “The Beginnings of a Probability Calculus,” inStudies in the History of Statistics and Probability, vol. 1, ed. E. S. Pearson and Sir Maurice Kendall (London: Charles Griffin & Co., 1970), 30.
When we can see the pattern of the losses and/or gains experienced in the past, we hope that the same pattern will continue in the future. In some cases, we want to be able to modify the past results in a logical way like inflating them for the time value of money discussed inChapter 4 "Evolving Risk Management: Fundamental Tools". If the patterns of gains and losses continue, our predictions of future losses or gains will be informative. Similarly, we may develop a pattern of losses based on theoretical or physical constructs (such as hurricane forecasting models based on physics or likelihood of obtaining a head in a flip of a coin based on theoretical models of equal likelihood of a head and a tail). Likelihood is the notion of how often a certain event will occur. Inaccuracies in our abilities to create a correct distribution arise from our inability to predict futures outcomes accurately. The distribution is the display of the events on a map that tells us the likelihood that the event or events will occur. In some ways, it resembles a picture of the likelihood and regularity of events that occur. Let’s now turn to creating models and measures of the outcomes and their frequency. Measures of Frequency and Severity Table 2.1 "Claims and Fire Losses for Group of Homes in Location A"and Table 2.2 "Claims and Fire Losses ($) for Homes in Location B"show the compilation of the number of claims and their dollar amounts for homes that were burnt during a five-year period in two different locations labeled Location A and Location B. We have information about the total number of claims per year and the amount of the fire losses in dollars for each year. Each location has the same number of homes (1,000 homes). Each location has a total of 51 claims for the five-year period, an average (or mean) of 10.2 claims per year, which is the frequency. The average dollar amount of losses per claim for the whole period is also the same for each location, $6,166.67, which is the definition of severity. Table 2.1 Claims and Fire Losses for Group of Homes in Location A
Table 2.2 Claims and Fire Losses ($) for Homes in Location B
As shown in Table 2.1 "Claims and Fire Losses for Group of Homes in Location A" and Table 2.2 "Claims and Fire Losses ($) for Homes in Location B", the total number of fire claims for the two locations A and B is the same, as is the total dollar amount of losses shown. You might recall from earlier, the number of claims per year is called the frequency. The average frequency of claims for locations A and B is 10.2 per year. The size of the loss in terms of dollars lost per claim is called severity, as we noted previously. The average dollars lost per claim per year in each location is $6,166.67. The most important measures for risk managers when they address potential losses that arise from uncertainty are usually those associated with frequency and severity of losses during a specified period of time. The use of frequency and severity data is very important to both insurers and firm managers concerned with judging the risk of various endeavors. Risk managers try to employ activities (physical construction, backup systems, financial hedging, insurance, etc.) to decrease the frequency or severity (or both) of potential losses. InChapter 4 "Evolving Risk Management: Fundamental Tools", we will see frequency data and severity data represented. Typically, the risk manager will relate the number of incidents under investigation to a base, such as the number of employees if examining the frequency and severity of workplace injuries. In the examples in Table 2.1 "Claims and Fire Losses for Group of Homes in Location A" and Table 2.2 "Claims and Fire Losses ($) for Homes in Location B", the severity is related to the number of fire claims in the five-year period per 1,000 homes. It is important to note that in these tables the precise distribution (frequencies and dollar losses) over the years for the claims per year arising in Location A is different from distribution for Location B. This will be discussed later in this chapter. Next, we discuss the concept of frequency in terms of probability or likelihood. Frequency and Probability Returning back to the quantification of the notion of uncertainty, we first observe that our intuitive usage of the word probability can have two different meanings or forms as related to statements of uncertain outcomes. This is exemplified by two different statements: [2]
Conceptually, these represent two distinct types of probability statements. The first is a statement about probability as a degree of belief about whether an event will occur and how firmly this belief is held. The second is a statement about how often a head would be expected to show up in repeated flips of a coin. The important difference is that the first statement’s validity or truth will be stated. We can clear up the statement’s veracity for all by sailing across the globe. The second statement, however, still remains unsettled. Even after the first coin flip, we still have a 50 percent chance that the next flip will result in a head. The second provides a different interpretation of “probability,” namely, as a relative frequency of occurrence in repeated trials. This relative frequency conceptualization of probability is most relevant for risk management. One wants to learn from past events about the likelihood of future occurrences. The discoverers of probability theory adopted the relative frequency approach to formalizing the likelihood of chance events. Pascal and Fermat ushered in a major conceptual breakthrough: the concept that, in repeated games of chance (or in many other situations encountered in nature) involving uncertainty, fixed relative frequencies of occurrence of the individual possible outcomes arose. These relative frequencies were both stable over time and individuals could calculate them by simply counting the number of ways that the outcome could occur divided by the total number of equally likely possible outcomes. In addition, empirically the relative frequency of occurrence of events in a long sequence of repeated trials (e.g., repeated gambling games) corresponded with the theoretical calculation of the number of ways an event could occur divided by the total number of possible outcomes. This is the model of equally likely outcomes or relative frequency definition of probability. It was a very distinct departure from the previous conceptualization of uncertainty that had all events controlled by God with no humanly discernable pattern. In the Pascal-Fermat framework, prediction became a matter of counting that could be done by anyone. Probability and prediction had become a tool of the people!Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up" provides an example representing all possible outcomes in the throw of two colored dice along with their associated probabilities. Figure 2.2 Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up 
Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up" lists the probabilities for the number of dots facing upward (2, 3, 4, etc.) in a roll of two colored dice. We can calculate the probability for any one of these numbers (2, 3, 4, etc.) by adding up the number of outcomes (rolls of two dice) that result in this number of dots facing up divided by the total number of possibilities. For example, a roll of thirty-six possibilities total when we roll two dice (count them). The probability of rolling a 2 is 1/36 (we can only roll a 2 one way, namely, when both dice have a 1 facing up). The probability of rolling a 7 is 6/36 = 1/6 (since rolls can fall any of six ways to roll a 7—1 and 6 twice, 2 and 5 twice, 3 and 4 twice). For any other choice of number of dots facing upward, we can get the probability by just adding the number of ways the event can occur divided by thirty-six. The probability of rolling a 7 or an 11 (5 and 6 twice) on a throw of the dice, for instance, is (6 + 2)/36 = 2/9. The notions of “equally likely outcomes” and the calculation of probabilities as the ratio of “the number of ways in which an event could occur, divided by the total number of equally likely outcomes” is seminal and instructive. But, it did not include situations in which the number of possible outcomes was (at least conceptually) unbounded or infinite or not equally likely. [3] We needed an extension. Noticing that the probability of an event, any event, provided that extension. Further, extending the theory to nonequally likely possible outcomes arose by noticing that the probability of an event—any event—occurring could be calculated as the relative frequency of an event occurring in a long run of trials in which the event may or may not occur. Thus, different events could have different, nonequal chances of occurring in a long repetition of scenarios involving the possible occurrences of the events. Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry" provides an example of this. We can extend the theory yet further to a situation in which the number of possible outcomes is potentially infinite. But what about a situation in which no easily definable bound on the number of possible outcomes can be found? We can address this situation by again using the relative frequency interpretation of probability as well. When we have a continuum of possible outcomes (e.g., if an outcome is time, we can view it as a continuous variable outcome), then a curve of relative frequency is created. Thus, the probability of an outcome falling between two numbers x and y is the area under the frequency curve between x and y. The total area under the curve is one reflecting that it’s 100 percent certain that some outcome will occur. The so-called normal distribution or bell-shaped curve from statistics provides us with an example of such a continuous probability distribution curve. The bell-shaped curve represents a situation wherein a continuum of possible outcomes arises. Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project" provides such a bell-shaped curve for the profitability of implementing a new research and development project. It may have profit or loss. Figure 2.3 Normal Distribution of Potential Profit from a Research and Development Project 
To find the probability of any range of profitability values for this research and development project, we find the area under the curve inFigure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project" between the desired range of profitability values. For example, the distribution in Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project" was constructed to have what is called a normal distribution with the hump over the point $30 million and a measure of spread of $23 million. This spread represents the standard deviation that we will discuss in the next section. We can calculate the area under the curve above $0, which will be the probability that we will make a profit by implementing the research and development project. We do this by reference to a normal distribution table of values available in any statistics book. The area under the curve is 0.904, meaning that we have approximately a 90 percent change (probability of 0.9) that the project will result in a profit. In practice, we build probability distribution tables or probability curves such as those in Figure 2.2 "Possible Outcomes for a Roll of Two Dice with the Probability of Having a Particular Number of Dots Facing Up", Figure 2.3 "Normal Distribution of Potential Profit from a Research and Development Project", and Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry" using estimates of the likelihood (probability) of various different states of nature based on either historical relative frequency of occurrence or theoretical data. For example, empirical data may come from repeated observations in similar situations such as with historically constructed life or mortality tables. Theoretical data may come from a physics or engineering assessment of failure likelihood for a bridge or nuclear power plant containment vessel. In some situations, however, we can determine the likelihoods subjectively or by expert opinion. For example, assessments of political overthrows of governments are used for pricing political risk insurance needed by corporations doing business in emerging markets. Regardless of the source of the likelihoods, we can obtain an assessment of the probabilities or relative frequencies of the future occurrence of each conceivable event. The resulting collection of possible events together with their respective probabilities of occurrence is called a probability distribution, an example of which is shown in Table 2.3 "Opportunity and Loss Assessment Consequences of New Product Market Entry". Directory: site -> textbooks textbooks -> This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License without attribution as requested by the work’s original creator or licensee. Preface textbooks -> Chapter 1 Introduction to Law textbooks -> 1. 1 Why Launch! textbooks -> This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License without attribution as requested by the work’s original creator or licensee textbooks -> This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License textbooks -> This text was adapted by The Saylor Foundation under a textbooks -> This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License without attribution as requested by the work’s original creator or licensee. Preface textbooks -> This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License textbooks -> Chapter 1 What Is Economics? textbooks -> This text was adapted by The Saylor Foundation under a Creative Commons Attribution-NonCommercial-ShareAlike 0 License Download 5.93 Mb. Share with your friends:
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